Direct collision with energy loss

1. A particle \(A\) has mass \(4 m\) and a particle \(B\) has mass \(3 m\). The particles are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(3 u\).
The direction of motion of each particle is reversed by the collision.
The total kinetic energy lost in the collision is \(\frac { 473 } { 24 } m u ^ { 2 }\)
Find

1. the coefficient of restitution between \(A\) and \(B\),
2. the magnitude of the impulse received by \(A\) in the collision.
   \section*{\textbackslash section*\{Question 8 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-28_2642_1833_118_118}

5. Two particles \(P\) and \(Q\), of masses \(2 m\) and \(m\) respectively, are on a smooth horizontal table. Particle \(Q\) is at rest and particle \(P\) collides directly with it when moving with speed \(u\). After the collision the total kinetic energy of the two particles is \(\frac { 3 } { 4 } m u ^ { 2 }\). Find

1. the speed of \(Q\) immediately after the collision,
2. the coefficient of restitution between the particles.

1 Two smooth spheres \(A\) and \(B\), of equal radii and of masses \(3 m\) and \(6 m\) respectively, are at rest on a smooth horizontal surface. Sphere \(A\) is projected directly towards \(B\) with speed \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Show that the kinetic energy lost in the collision between \(A\) and \(B\) is \(m u ^ { 2 } \left( 1 - e ^ { 2 } \right)\).

3 Two uniform small smooth spheres \(A\) and \(B\), of masses \(m\) and \(2 m\) respectively, and with equal radii, are at rest on a smooth horizontal surface. Sphere \(A\) is projected directly towards \(B\) with speed \(u\), and collides with \(B\). After this collision, sphere \(B\) collides directly with a fixed smooth vertical barrier. The total kinetic energy of the spheres after this second collision is equal to one-ninth of its value before the first collision. Given that the coefficient of restitution between \(B\) and the barrier is 0.5 , find the coefficient of restitution between \(A\) and \(B\).

2 A small smooth sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with an identical sphere \(B\) which is initially at rest on the surface. The coefficient of restitution between the spheres is \(e\). Sphere \(B\) subsequently collides with a fixed vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac { 1 } { 2 }\). Given that \(80 \%\) of the initial kinetic energy is lost as a result of the two collisions, find the value of \(e\).

5 Two spheres of the same radius with masses 2 kg and 3 kg are moving directly towards each other on a smooth horizontal plane with speeds \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The spheres collide and the kinetic energy lost is 81 J . Calculate the speed and direction of motion of each sphere after the collision.

1 Two stones, A and B , are sliding along the same straight line on a horizontal sheet of ice. Stone A, of mass 50 kg , is moving with a constant velocity of \(2.1 \mathrm {~ms} ^ { - 1 }\) towards stone B. Stone B, of mass 70 kg , is moving with a constant velocity of \(0.8 \mathrm {~ms} ^ { - 1 }\) towards stone A. A and B collide directly. Immediately after their collision stone A's velocity is \(0.35 \mathrm {~ms} ^ { - 1 }\) in the same direction as its velocity before the collision.

1. Find the speed of stone B immediately after the collision.
2. Find the coefficient of restitution for the collision.
3. Find the total loss of kinetic energy caused by the collision.
4. Explain whether the collision was perfectly elastic.

6 Two small spheres \(A\) and \(B\), of masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. \(A\) has speed \(4 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) before they collide. The coefficient of restitution between \(A\) and \(B\) is 0.4 .

1. Find the speed of each sphere after the collision.
2. Find, in terms of \(m\), the loss of kinetic energy during the collision.
3. Given that the magnitude of the impulse exerted on \(A\) by \(B\) during the collision is 2.52 Ns , find \(m\).

4

1. Two discs, P of mass 4 kg and Q of mass 5 kg , are sliding along the same line on a smooth horizontal plane when they collide. The velocity of P before the collision and the velocity of Q after the collision are shown in Fig. 4. P loses \(\frac { 5 } { 9 }\) of its kinetic energy in the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{71d839d8-12ca-4806-8f74-c572e7e21891-5_294_899_390_584} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Show that after the collision P has a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the opposite direction to its original motion. While colliding, the discs are in contact for \(\frac { 1 } { 5 } \mathrm {~s}\).
   2. Find the impulse on P in the collision and the average force acting on the discs.
   3. Find the velocity of Q before the collision and the coefficient of restitution between the two discs.
2. A particle is projected from a point 2.5 m above a smooth horizontal plane. Its initial velocity is \(5.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) below the horizontal, where \(\sin \theta = \frac { 15 } { 17 }\). The coefficient of restitution between the particle and the plane is \(\frac { 4 } { 5 }\).
   1. Show that, after bouncing off the plane, the greatest height reached by the particle is 2.5 m .
   2. Calculate the horizontal distance between the two points at which the particle is 2.5 m above the plane.

4 Two model railway trucks, A of mass 0.1 kg and B of mass 0.2 kg , are constrained to move on a smooth straight level track.
Initially B is stationary and A is moving towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before they collide. The coefficient of restitution between A and B is \(e\).

1. Find the speed of A and the speed of B after the collision, giving your answers in terms of \(e\) and \(u\).
2. Show that the loss of kinetic energy in the collision is \(\frac { 1 } { 30 } u ^ { 2 } \left( 1 - e ^ { 2 } \right)\).
3. For the case in which the loss of kinetic energy is least
   • state the value of \(e\)
   • state the loss in kinetic energy
   • describe the subsequent motion of the trucks.
   • For the case in which the loss of kinetic energy is greatest
   • state the value of \(e\)
   • state the loss in kinetic energy
   • describe the subsequent motion of the trucks.

4 Fig. 4.1 shows two spheres, A and B, on a smooth horizontal surface. Their masses are 3 kg and 1 kg respectively. \begin{figure}[h] \end{figure} Initially, sphere A travels at a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line towards B , which is at rest. The spheres collide and the coefficient of restitution between A and B is \(e\).

1. Show that, after the collision, A has a speed of \(\frac { 1 } { 4 } ( 3 - e ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and find an expression for the speed of B in terms of \(e\). During the collision, the kinetic energy of the system decreases by \(21 \%\).
2. Determine the value of \(e\).
3. State why in part (a) it was necessary to assume that A and B have equal radii. Fig. 4.2 shows two spheres, C and D , of equal radii on a smooth horizontal surface. Their masses are 1 kg and 2 kg respectively. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 4.2} \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-4_158_1155_1544_244}
   \end{figure} Spheres C and D travel towards each other along the same straight line, C with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and D with a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The spheres collide and during the collision C exerts an impulse on D of magnitude \(\frac { 2 } { 3 } ( u + 1 ) \mathrm { Ns }\).
4. Show that C and D have the same velocity after the collision.
5. Determine the fraction of kinetic energy lost due to the collision between C and D as \(u \rightarrow \infty\).

5. The diagram below shows two spheres \(A\) and \(B\), of equal radii, moving in the same direction on a smooth horizontal surface. Sphere \(A\), of mass 3 kg , is moving with speed \(4 \mathrm {~ms} ^ { - 1 }\) and sphere \(B\), of mass 2 kg , is moving with speed \(10 \mathrm {~ms} ^ { - 1 }\).
\includegraphics[max width=\textwidth, alt={}, center]{b6b77d92-acb6-4cc3-8328-1899ed4a87cd-6_275_972_580_548} Sphere \(B\) is then given an impulse after which it moves in the opposite direction with speed \(6 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the magnitude and direction of the impulse exerted on \(B\). Sphere \(B\) continues to move with speed \(6 \mathrm {~ms} ^ { - 1 }\) so that it collides directly with sphere \(A\). The kinetic energy lost due to the collision is 45 J .
2. Calculate the speed of \(A\) and the speed of \(B\) immediately after the two spheres collide. State the direction in which each sphere is moving relative to its motion immediately before the collision.

1. A particle \(P\) of mass \(3 m\) and a particle \(Q\) of mass \(2 m\) are moving along the same straight line on a smooth horizontal plane. The particles are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2 u\).
Immediately after the collision \(P\) and \(Q\) are moving in opposite directions.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the range of possible values of \(e\), justifying your answer. Given that \(Q\) loses 75\% of its kinetic energy as a result of the collision,
2. find the value of \(e\).

1. Two particles, \(A\) and \(B\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.

Particle \(A\) has mass \(5 m\) and particle \(B\) has mass \(3 m\).
The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\)
Immediately after the collision the speed of \(A\) is \(v\) and the speed of \(B\) is \(2 v\).
Given that \(A\) and \(B\) are moving in the same direction after the collision,

1. find the set of possible values of \(e\). Given also that the kinetic energy of \(A\) immediately after the collision is \(16 \%\) of the kinetic energy of \(A\) immediately before the collision,
2. find
   1. the value of \(e\),
   2. the magnitude of the impulse received by \(A\) in the collision, giving your answer in terms of \(m\) and \(v\).

7 The masses of two particles \(A\) and \(B\) are \(m\) and \(2 m\) respectively. They are moving towards each other on a smooth horizontal table. Just before they collide their speeds are \(u\) and \(2 u\) respectively. After the collision the kinetic energy of \(A\) is 8 times the kinetic energy of \(B\). Find the coefficient of restitution between \(A\) and \(B\). \section*{END OF QUESTION PAPER}